The intersection of a plane with a right circular cylinder could be which of the following ?
I. A circle
II . Parallel lines
III. Intersecting lines

Options

A. I only
B. II only
C. III only
D. I and II only

Explanation

To determine the possible intersections of a plane with a right circular cylinder, we will analyze each option provided in the question.

### Step 1: Understanding the Geometry
A right circular cylinder has a circular base and extends vertically. When a plane intersects with this cylinder, the shape of the intersection depends on the orientation of the plane.

**Hint:** Visualize the cylinder and consider how a flat surface (the plane) can cut through it.

### Step 2: Analyzing Option I - A Circle
If the plane is horizontal and cuts through the cylinder, the intersection will be a circle. This is because the cross-section of the cylinder at any height is a circle.

**Hint:** Think about how slicing a cylindrical object horizontally yields a circular shape.

### Step 3: Analyzing Option II - Parallel Lines
If the plane cuts vertically and is parallel to the axis of the cylinder, the intersection will yield two parallel lines. This occurs because the plane intersects the curved surface of the cylinder at two points, creating two parallel lines.

**Hint:** Imagine a vertical slice through the cylinder that does not touch the top or bottom edges.

### Step 4: Analyzing Option III - Intersecting Lines
For the intersection to yield intersecting lines, the plane would need to cut through the cylinder at an angle that creates two lines that cross each other. However, this is not possible with a right circular cylinder because the surface is curved. Therefore, intersecting lines cannot be formed.

**Hint:** Consider the nature of the cylinder's surface and how a plane interacts with it.

### Conclusion
Based on the analysis:
- A circle (Option I) is possible.
- Parallel lines (Option II) are also possible.
- Intersecting lines (Option III) are not possible.

Thus, the correct answer is that the intersection can be a circle or parallel lines, which corresponds to option 4 (One and Two only).

**Final Answer:** Option 4 (One and Two only).